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\input amssym.def            % small letters for UNIX,  not: AMSsym.def
\input epsf.def% \input epsf %for UNIX
%\input epsf          %\input epsf.def for MAC f"ur BILDER!!
\input pics.tex

\input BoxedEPS
\SetTexturesEPSFSpecial
\HideDisplacementBoxes

\def\lf{\ \hfil\break}       % Neue Zeile ohne Einr"ucken, 'linefeed'
\def\cl{\centerline}
\def\Lf{\vskip1pt\noindent}   % Neue Zeile mit breiterem Zwischenraum
\def\LF{\medskip\noindent}   % Neue Zeile mit breiterem Zwischenraum
\def\R90{{\rm Rot}(90^\circ)}
\def\Dd#1{{\partial \over \partial #1}}

\nopagenumbers

\vglue -10pt
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\cl {\cmrX  About  The Feigenbaum Tree }
\lf
\cl { See also: Julia Set of $z\to (z^2-c)$  }
\Lf
The Feigenbaum Tree is one of the earliest examples of parameter
dependent behavior of a dynamical system. 
The dynamical system in question is called the {\it Logistic Map}:
$$ f_\mu(y) := 4\mu \cdot y(1-y), \  y\in [0,1],\  \mu\in [1/4,1].$$
Since both the parameter space, $ [1/4,1]$, and the dynamical space, $ [0,1]$, 
are 1-dimensional, 
one can illustrate in a $(\mu,y)$-plane how the dynamical behavior changes
as the parameter $\mu$ varies. The usual experiment (and the one used in 3DXM) 
goes as follows: Starting with a set of initial values $\{y_k;\ y_k\in[0,1],\ k=1,\dots,K\}$
(and with as many parameter values $\mu$ as one wants to handle) one computes
many iterations $f^{\circ n}_\mu(y_k), n =1,\dots,N$ with $N$ large.
\Lf  If one plots
only the iterations with say $n\ge 1000$, then one observes in the $(\mu,y)$-plane
the {\it Feigenbaum Tree}: for small $\mu$ the iterated points $f^{\circ n}_\mu(y_k)$
converge to a stable fixed point of the map $f_\mu$, $y_f = f_\mu(y_f),\ y_f:=1-1/4\mu$. 
Observe that the derivative $f'$ at the fixed point is $2-4\mu\le 0$. At $\mu=3/4$ the
derivative at the fixed point is $-1$, so that the fixed point stops being attractive. It
turns out that for larger $\mu$ the orbit of period 2 is attractive for a while -- until
$\mu$ reaches another bifurcation point after which an orbit of period 4 becomes
attractive.
\Lf
This period doubling ``cascade'' continues up to a certain $\mu$-value, past which 
there is for a while no longer an attractive orbit. All this is clearly visible in the
3DXM demo. One should use the Action Menu entry: {\it Iterate Mouse Point Forward}
to watch how arbitrary initial points are iterated and how these iterations converge
to the attracting orbits of period $2^d$ in the left, period doubling, part of the
Feigenbaum Tree. ---{\it Speed-Up Note: } If one presses DELETE either during
the default iterations or during the iteration of a point chosen by mouse, then all
delays are skipped and the result of the iteration is reached much more quickly.
\Lf
After the period doubling in the left part has been observed one wants to look at the right 
part of the Feigenbaum Tree more closely. The $\mu$-interval which the illustration uses
is the interval $[bb, cc]$. It can be changed in the Parameter  entry of the Settings Menu.
Since the attractive orbit of period 2 appears after $\mu=0.75$, one loses only the
simple attractors if one increases $bb$ from $0.25$ to $0.75$, and one gains that the remaining 
part of the Tree is stretched by a factor of $3$. In the same way one can magnify any part
of the parameter space. Of course the dynamical space is always fully shown---unless
one decides to use SHIFT+MOUSE to scale the image to see part of the dynamical
space magnified. In this case translation using CONTROL+MOUSE-DRAG may be useful.
\Lf
The most obvious feature in the right part of the Feigenbaum Tree are gaps, three fairly
large ones and any number of thinner ones. The three large ones belong to parameter
intervals where the map $f_\mu$ has attractive orbits of period 6, period 5, resp. period 3. 
If one magnifies a gap enough, one can experimentally check that the gaps belong to
attractive orbits (use in the Action Menu {\it Iterate Mouse Point Forward}). One also observes
that at the right end of these intervals the periods double again, and again. In other words,
the Feigenbaum Tree illuminates, almost at the first glimpse, many properties of this
1-parameter family of iterated maps.
\Lf
Finally we remark that the Feigenbaum Tree is related to the real part of the Mandelbrot
set because the Mandelbrot set also parametrizes quadratic maps $z \to f_c(z):=(z^2-c)$
according to their dynamical properties. If $c$ is chosen from the big bottom apple then
$f_c$ has an attractive fixed point. As one passes on the real axis from the apple to the disk
above it, the fixed point changes from attractive through indifferent to unstable and the
orbit of period 2 becomes attractive. As one moves (always along the real axis) towards
the top of the Mandelbrot set one continues to meet exactly the same kind of dynamical
behavior as one sees in the Feigenbaum Tree. For more details see the documentation
for {\it Julia Set of $z\to (z^2-c)$}.
\lf

\bye
LogisticMap := 4 * mu * y * (1 - y); 

 